1904.] 



BROOES — OETHIC CURVES. 311 



There are, in general, six real foci, which fall into two sets of 

 three. Each set of three corresponds to a single poiiit of the z-pla?ie, 

 and is, therefore, a maxitnum inscribed triangle of one of the ellipses 

 described above. 



XV. The Foci and the Branch Points. 

 If we eliminate the parameter between 



22 = a^ -f- ^^\ 

 and 



we get the equation of the line ^, 



a^z — a^z = a^a^ — a^a^. 



Now suppose, for a moment, that this line does not contain either 

 of the branch points z=^^\. Tlien, if we put 2; = ±i in the 

 equation of the line and solve for z, we get a value which is not the con- 

 jugate of 5, but is the vector of the reflection of the point 2 = rfc i 

 in the line considered. The three points in the .r-plane got by 

 putting 



in the equation 



x^ — S-^ = 22 



are the points mapped in the s-plane by the reflection of 2 = ± i in 

 the line ?. It follows that : 



The real foci of the orthic cubic which corresponds to a given line 

 are the six points which correspond to the reflections in that line of 

 the branch points. 



If the director line pass through one of the branch points {i.e., if 



= — is real), two foci coincide to form the node, and the 



remaining one of the set is on the curve. One who looks at the 

 matter from the point of view of the Riemann surface might be 

 surprised that a branch point is to be reflected in the line in each 

 sheet of the surface and not in the two sheets alone which it con- 

 nects. A moment's consideration will show that whether or not 



